Glossary

7.3 Summation of series

4 min read•august 13, 2024

offers a powerful method for summing complex series. By expressing series terms as coefficients of Laurent expansions for meromorphic functions, we can leverage contour integrals and the to find closed-form solutions.

This approach is particularly useful for , , and series involving rational functions. We'll explore how to identify suitable series, use generating functions, and apply residue calculations to solve challenging summation problems.

Series Summation with Residues

Identifying Series Suitable for Residue Theory

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  • Residue theory is applicable when series terms can be expressed as coefficients of a for a meromorphic function
  • Power series, trigonometric series (Fourier series), and series involving rational functions are commonly summed using residue theory
  • The of a series is a formal power series with coefficients equal to the terms of the original series
    • Generating functions can often be expressed as complex functions
  • Residue theorem is applicable if the generating function has poles ()

Generating Functions and Complex Integrals

  • Generating functions express series as complex contour integrals
  • Contour integrals take the form ∮f(z)g(z)dz\oint f(z) g(z) dz
    • f(z)f(z) represents the generating function
    • g(z)g(z) is a suitable function chosen to extract desired series coefficients
  • Contours are typically closed curves enclosing the origin (circles or )
  • The choice of g(z)g(z) depends on the specific series and desired result form
    • 1/zn+11/z^{n+1} is common for power series
    • 1/(e2πiz−1)1/(e^{2\pi iz} - 1) is common for trigonometric series

Series as Complex Integrals

Residue Theorem Application

  • The residue theorem equates the of a meromorphic function f(z)f(z) over a CC to 2πi2\pi i times the sum of the residues of f(z)f(z) at its poles inside CC
  • Applying the residue theorem involves identifying the poles of the integrand (product of generating function and chosen g(z)g(z)) inside the contour
  • Residues are calculated at each pole using appropriate methods
    • Formula for
    • Laurent series expansion for
  • Summing the residues and multiplying by 2πi2\pi i yields the contour integral value

Contour Integral Evaluation

  • The contour integral result obtained using the residue theorem equals the sum of the original series
  • Series coefficients are extracted from the contour integral result to obtain a closed-form expression for the series sum
  • The closed-form expression may involve well-known constants (π\pi, ee) or special functions (, )
  • Simplifying the closed-form expression leads to a more concise or insightful representation of the series sum

Evaluating Complex Integrals with Residues

Residue Calculation Methods

  • Simple poles: The residue at a simple pole z0z_0 of a function f(z)f(z) is given by lim⁡z→z0(z−z0)f(z)\lim_{z \to z_0} (z - z_0) f(z)
  • Higher-order poles: For a pole of order nn at z0z_0, the residue is the coefficient of (z−z0)−1(z - z_0)^{-1} in the Laurent series expansion of f(z)f(z) around z0z_0
    • The coefficient can be found using the formula 1(n−1)!lim⁡z→z0dn−1dzn−1((z−z0)nf(z))\frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}} ((z - z_0)^n f(z))
  • : If f(z)f(z) has a pole at infinity, the residue can be calculated by considering the residue at zero of the function f(1/z)/z2f(1/z)/z^2

Contour Selection and Deformation

  • The choice of contour depends on the location of the poles and the desired form of the result
  • Circular contours are commonly used for series with poles at the origin or evenly spaced poles
  • Keyhole contours (indented circles) are useful for series with poles near the origin or on the real axis
  • techniques, such as indenting the contour or using branch cuts, are employed to avoid singularities or to simplify the integral

Closed-Form Expressions for Series Sums

Interpreting Contour Integral Results

  • The contour integral result represents the sum of the original series
  • Extracting series coefficients from the contour integral result leads to a closed-form expression for the series sum
  • The closed-form expression may involve constants, special functions, or finite sums
    • Examples of constants: π\pi, ee, (γ\gamma)
    • Examples of special functions: Bernoulli numbers (BnB_n), polygamma functions (ψ(n)(z)\psi^{(n)}(z))

Verifying and Applying Series Summation Results

  • Closed-form expressions can be simplified to obtain more concise or insightful representations of the series sum
  • Results can be verified by comparing with known series summation formulas or numerically evaluating the series for specific parameter values
  • Series summation results have applications in various fields
    • Evaluating infinite sums in mathematical analysis
    • Deriving closed-form expressions for special functions
    • Solving problems in physics and engineering (Fourier series, generating functions)

Key Terms to Review (18)

Bernoulli numbers: Bernoulli numbers are a sequence of rational numbers that are deeply connected to number theory and mathematical analysis, particularly in the study of summation of series and special functions like the gamma and zeta functions. They play a crucial role in the calculation of power sums, generating functions, and the expansion of certain functions into Taylor series. Their significance extends to various mathematical fields, showcasing their utility in approximating sums and understanding properties of polynomial sequences.
Closed contour: A closed contour is a continuous curve in the complex plane that starts and ends at the same point, effectively enclosing a region. It plays a vital role in various applications such as calculating integrals, analyzing series, and understanding the properties of meromorphic functions. Closed contours are essential for applying the residue theorem and other key concepts in complex analysis.
Contour Deformation: Contour deformation refers to the process of altering the path of a contour integral without changing the value of the integral, as long as certain conditions are met, such as the integrand being analytic within the region bounded by the original and new contours. This concept is fundamental in complex analysis, especially when dealing with the evaluation of integrals and summation of series, as it allows for simplification and manipulation of integrals to obtain desired results.
Contour Integral: A contour integral is a type of integral that evaluates a complex-valued function along a specified path or contour in the complex plane. This method allows for the evaluation of integrals that are not easily computed using traditional methods, by leveraging properties of complex functions such as analyticity and the behavior of singularities.
Euler's Constant: Euler's constant, often denoted as $$ ext{e}$$, is a fundamental mathematical constant approximately equal to 2.71828. It serves as the base of natural logarithms and has vital applications in various fields, especially in calculus and complex analysis. This constant emerges naturally in various contexts, such as in the study of exponential growth and decay, making it essential for understanding many series and limits.
Generating Function: A generating function is a formal power series in one variable that encodes a sequence of coefficients, often used to represent and manipulate sequences and series in mathematics. It serves as a tool for deriving properties of the sequence, facilitating operations like addition, multiplication, and finding closed forms for series. Generating functions can simplify complex combinatorial problems and help solve recurrence relations by translating them into algebraic equations.
Higher-order poles: Higher-order poles refer to singularities of a meromorphic function where the pole has an order greater than one. This means that the function can be expressed in the form of a Laurent series, featuring negative powers of $(z - z_0)$ where $z_0$ is the location of the pole. Understanding higher-order poles is crucial for analyzing the behavior of complex functions near their singular points, especially in relation to residues and contour integration.
Isolated Singularities: Isolated singularities are points in the complex plane where a function ceases to be analytic, but they are surrounded by a neighborhood in which the function is analytic. These singularities can be classified as removable, poles, or essential, each influencing how the function behaves near these points. Understanding isolated singularities is crucial for evaluating integrals and summing series involving complex functions.
Keyhole Contours: Keyhole contours are specific types of integration paths used in complex analysis that resemble the shape of a keyhole. They are particularly useful for evaluating integrals that involve singularities or branch points, allowing the integral to be computed around a contour that avoids these problematic areas while capturing essential contributions from the residue at the singularity.
Laurent series: A Laurent series is a representation of a complex function as a series that includes both positive and negative powers of the variable, typically centered around a singularity. This series provides insights into the behavior of complex functions in regions that include singular points, allowing for the analysis of their properties such as convergence and residues.
Polygamma Functions: Polygamma functions are a set of special functions that are defined as the derivatives of the logarithm of the gamma function. They play an important role in summation of series and can be used to express various infinite series and sequences in terms of more manageable forms. The polygamma function is denoted as \( \psi^{(n)}(x) \), where \( n \) indicates the order of the derivative, with \( \psi^{(0)}(x) \) being the digamma function, which is the first derivative of the logarithm of the gamma function.
Power Series: A power series is an infinite series of the form $$ ext{f}(z) = ext{a}_0 + ext{a}_1 z + ext{a}_2 z^2 + ext{a}_3 z^3 + ...$$, where $$ ext{a}_n$$ are complex coefficients and $$z$$ is a complex variable. This type of series can represent analytic functions within a certain radius of convergence, connecting the concepts of sequences, analytic functions, and the broader realm of complex analysis.
Residue calculation: Residue calculation is a method in complex analysis used to find the value of certain types of integrals, particularly those involving singularities. It is based on the residue theorem, which states that the integral of a function around a closed contour can be evaluated using the residues of its poles inside the contour. This technique simplifies the evaluation of integrals and is essential in connecting complex analysis to various applications, including summation of series.
Residue Theorem: The residue theorem is a powerful result in complex analysis that relates contour integrals of holomorphic functions around singularities to the residues at those singularities. It states that the integral of a function over a closed contour can be calculated by summing the residues of the function's singular points enclosed by the contour, multiplied by $2\pi i$. This theorem serves as a cornerstone for evaluating integrals and series in complex analysis and has broad applications in real integrals, physics, and engineering.
Residue Theory: Residue theory is a powerful tool in complex analysis that deals with the evaluation of integrals and the summation of series by analyzing singularities of analytic functions. It leverages the concept of residues, which are coefficients in the Laurent series expansion of a function around its singular points. This approach simplifies complex calculations by transforming difficult contour integrals into manageable sums involving residues at poles.
Residues at Infinity: Residues at infinity are the values that represent the behavior of a complex function as it approaches infinity. In the context of complex analysis, they help in evaluating integrals and understanding the overall behavior of functions by relating them to residues at finite poles through the residue theorem. This concept is crucial when summing series, especially when considering contributions from terms that may not converge as expected.
Simple Poles: Simple poles are specific types of singularities in complex analysis where a function behaves like \\frac{1}{z-a} near the pole, for some point \(a\). They are characterized by having a residue that is non-zero, which allows us to use them in contour integrals and series representations. Understanding simple poles is crucial because they play a significant role in the behavior of meromorphic functions and in the evaluation of integrals through residues.
Trigonometric Series: A trigonometric series is a series of the form $$ ext{a}_0 + ext{a}_1 ext{cos}(nx) + ext{b}_1 ext{sin}(nx) + ext{a}_2 ext{cos}(2nx) + ext{b}_2 ext{sin}(2nx) + ...$$ where the coefficients are constants and the terms involve sine and cosine functions. These series are often used to represent periodic functions and can converge to a function in certain intervals. They play a crucial role in Fourier analysis and can be used to analyze and reconstruct signals in various fields.
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